Quantum-mechanics theories {matrix quantum mechanics} {S-matrix theory} can use linear-equation systems, with indexed terms, to model electronic-transition energies.
transition matrix
Square matrices can represent linear-equation systems. Infinite square matrices can represent Hilbert spaces with infinitely many dimensions. Matrix rows and columns represent the same energy levels. Matrices are infinite, because particles can go to any energy level, and energy levels can go higher infinitely. Matrix cells represent possible particle-energy-level transitions and their probabilities. Matrix elements are time-dependent complex numbers in infinite Hilbert space. Squared-amplitude absolute values give probabilities of energy-level transitions.
Matrix cells include all direct and cross-channel electronic transitions. Cells (linear-equation terms) with both indices the same are for directly emitted or absorbed photons. Cells (linear-equation terms) with different indices are for cross channels.
Because transition-matrix amplitudes are renormalized, sum of all state probabilities is one. Transition matrices are mathematically equivalent to Schrödinger wave equations, because time-dependent complex numbers represent anharmonic oscillators.
quanta
Matrix cells represent discrete energy changes and so quanta. Matrices are not continuous.
deterministic
Particles move from energy state to energy state deterministically, with probabilities.
space
Transition matrices are not about space. There is no position or trajectory information.
space: no fields
Energy and momentum transfers are quanta. There are no fields.
space: uncertainty
Matrices use non-commutative symbol algebra, not wave-equation Hamiltonian-equation variables. The uncertainty principle depends on wave behavior. Non-commuting operators are certain, so matrix theory does not account for uncertainty.
time
Transition matrices can change over time.
tensor
Quantum-mechanical matrices are similar to general-relativity symmetric tensors. Hermitean-matrix principal-axis transformation is a unitary-Hilbert-space tensor. If transformation is independent of time, tensor is a diagonal matrix. However, quadratic distance form is invariant, so transformations are unitary, not orthogonal as in general relativity.
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Date Modified: 2022.0224